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Theorem iunin1 3749
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3738 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
iunin1  |-  U_ x  e.  A  ( C  i^i  B )  =  (
U_ x  e.  A  C  i^i  B )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem iunin1
StepHypRef Expression
1 iunin2 3748 . 2  |-  U_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  U_ x  e.  A  C )
2 incom 3157 . . . 4  |-  ( C  i^i  B )  =  ( B  i^i  C
)
32a1i 9 . . 3  |-  ( x  e.  A  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
43iuneq2i 3703 . 2  |-  U_ x  e.  A  ( C  i^i  B )  =  U_ x  e.  A  ( B  i^i  C )
5 incom 3157 . 2  |-  ( U_ x  e.  A  C  i^i  B )  =  ( B  i^i  U_ x  e.  A  C )
61, 4, 53eqtr4i 2086 1  |-  U_ x  e.  A  ( C  i^i  B )  =  (
U_ x  e.  A  C  i^i  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409    i^i cin 2944   U_ciun 3685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-iun 3687
This theorem is referenced by:  2iunin  3751
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